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1. Crouzeix's conjecture, compressions of shifts, and classes of nilpotent matrices

   https://doi.org/10.1515/conop-2024-0004  

   Bickel, Kelly; Corbett, Georgia; Glenning, Annie; Guan, Changkun; Vollmayr-Lee, Martin (January 2024, Concrete Operators)

   Abstract This article studies the level set Crouzeix (LSC) conjecture, which is a weak version of Crouzeix’s conjecture that applies to finite compressions of the shift. Among other results, this article establishes the LSC conjecture for several classes of $3\times 3$3\times 3, $4\times 4$4\times 4, and $5\times 5$5\times 5matrices associated with compressions of the shift via a geometric analysis of their numerical ranges. This study also establishes Crouzeix’s conjecture for several classes of nilpotent matrices whose studies are motivated by related compressions of shifts. 

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2. Generalized Alder-Type Partition Inequalities

   https://doi.org/10.37236/11606  

   Armstrong, Liam; Ducasse, Bryan; Meyer, Thomas; Swisher, Holly (June 2023, The Electronic Journal of Combinatorics)

   In 2020, Kang and Park conjectured a "level $$2$$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $$3$$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $$n$$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities. 

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3. Restricted Tangent Bundles for General Free Rational Curves

   https://doi.org/10.1093/imrn/rnac011  

   Lehmann, Brian; Riedl, Eric (May 2022, International Mathematics Research Notices)

   Abstract Suppose that $$X$$ is a smooth projective variety and that $$C$$ is a general member of a family of free rational curves on $$X$$. We prove several statements showing that the Harder–Narasimhan filtration of $$T_{X}|_{C}$$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $$T_{X}$$ with respect to the class of $$C$$. When $$X$$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves. 

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4. KRW Composition Theorems via Lifting

   de Rezende, Susanna F; Meir, O; Nordström, J; Pitassi, T; Robere, R. (July 2020, Electronic colloquium on computational complexity)

   One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P 6⊆ NC1). Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions f ⋄ g. They showed that the validity of this conjecture would imply that P 6⊆ NC^1 . Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function f, but only for few inner functions g. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function g whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function f with the monotone complexity of the inner function g. In this setting, we prove the KRW conjecture for a similar selection of inner functions g, but only for a specific choice of the outer function f. 

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5. On Turán exponents of bipartite graphs

   https://doi.org/10.1017/S0963548321000341  

   Jiang, Tao; Ma, Jie; Yepremyan, Liana (March 2022, Combinatorics, Probability and Computing)

   Abstract A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $$r\in (1,2)$$ there exists a bipartite graph H such that $$\mathrm{ex}(n,H)=\Theta(n^r)$$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $$k\geq 2$$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $$k\geq 2$$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $$^{\prime}$$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $$^{\prime}$$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $$^{\prime}$$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ . 

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